Optimal. Leaf size=17 \[ x+\frac {1}{e^x+1}-\log \left (e^x+1\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2282, 44} \[ x+\frac {1}{e^x+1}-\log \left (e^x+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 2282
Rubi steps
\begin {align*} \int \frac {1}{1+2 e^x+e^{2 x}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{x (1+x)^2} \, dx,x,e^x\right )\\ &=\operatorname {Subst}\left (\int \left (\frac {1}{-1-x}+\frac {1}{x}-\frac {1}{(1+x)^2}\right ) \, dx,x,e^x\right )\\ &=\frac {1}{1+e^x}+x-\log \left (1+e^x\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 17, normalized size = 1.00 \[ x+\frac {1}{e^x+1}-\log \left (e^x+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 25, normalized size = 1.47 \[ \frac {x e^{x} - {\left (e^{x} + 1\right )} \log \left (e^{x} + 1\right ) + x + 1}{e^{x} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.24, size = 15, normalized size = 0.88 \[ x + \frac {1}{e^{x} + 1} - \log \left (e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 18, normalized size = 1.06 \[ -\ln \left ({\mathrm e}^{x}+1\right )+\ln \left ({\mathrm e}^{x}\right )+\frac {1}{{\mathrm e}^{x}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.59, size = 15, normalized size = 0.88 \[ x + \frac {1}{e^{x} + 1} - \log \left (e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.26, size = 15, normalized size = 0.88 \[ x-\ln \left ({\mathrm {e}}^x+1\right )+\frac {1}{{\mathrm {e}}^x+1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.09, size = 14, normalized size = 0.82 \[ x - \log {\left (e^{x} + 1 \right )} + \frac {1}{e^{x} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________